1.
ISSN: 2278 – 1323
International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 2, Issue 6, June 2013
2091
www.ijarcet.org
Optimization of Hamilton Path
Tournament Schedule1
Trinetra Kumar Pathak, 2
Vinod Kumar Bhalla
Abstract: A Hamilton path tournament design
involving n teams and n/2 stadiums, is a round robin
schedule on n − 1 days in which each team plays in
each stadium at most twice, and the set of games
played in each stadium induce a Hamilton path on n
teams[1]. We have created an algorithm to calculate
the cost of all pairs which is generated using Hamilton
path tournament Design [1] that work on n teams,
,n/2 stadiums and n − 1 days schedule in which each
team played at most twice in each stadium [1]. In our
algorithm, we have calculated the cost of each
combination of pairs and choose the Optimize
Hamilton path tournament.
Index Terms – Cost, Combination, Schedule, Path.
1 INTRODUCTION
There are number of algorithms for scheduling
balanced tournament designs [2], [6], [7]. Yoshiko
T. Ikebe and Akihisa Tamura [1] introduced a
Hamilton path tournament design and have proved
it for all n = 2p ≥ 8 teams. Gelling and Odeh
introduced the balanced tournament design [4], and
Schellenberg, van Rees and Vanstone proved its
existence for all even n ≥ 6 [9]. In this paper, we
have introduced cost effect approach for
tournament design using Hamilton path. We have
used cost constant to find the optimized
combination of pairs of balance tournament.
Several authors have recently have proposed a
schedule closely related to balanced tournament
design [3], [5], [8], [10]. We have designed an
optimized Hamilton path tournament in which there
are n teams, n-1 days and n/2 stadiums. Cost for
each and every team in each combination of pairs is
calculated and then add these costs in a single one
for each pair and compare with last minimum cost
and at last we got the optimize combination of
Hamilton path tournament.
II CONSTRUCTIONS
First, we have taken combination of pair in which
we have n team means n/2 stadium. Then, we
calculate the cost of each and every team traversal
according to a given combination of pair. Suppose
team t1 play first match in stadium s1 than we
initiate the cost of team t1 as 0 and check the next
match play by team t1, if t1 play second match in
stadium s2 the add the transportation cost of s1 to
s2 into cost of team t1 and check for the next match
for team t1 up to last match of t1 and same
procedure apply for the all remaining teams and
then add the cost of all teams which is called First
Combination Cost.
Now this procedure applies for all the combinations
of pairs and then we select the best Combination
of pairs generated by Hamilton path tournament.
II ILLUSTRATION USING EXAMPLE
Table 1 indicate the one combination of pairs
according to Hamilton path tournament for n=8.
Let there are 8 team t1, t2, t3, t4, t5, t6, t7 and t8, 4
stadiums s s1, s2, s3 and s4 and 7 days d1, d2, d3,
d4, d5, d6 and d7 [1].
Days
Stadium
s1 s2 s3 s4
d1 1,5 2,6 3,7 4,8
d2 1,6 5,7 2,8 3,4
d3 6,8 2,3 5,4 1,7
d4 7,4 6,3 1,8 5,2
d5 8,3 1,4 7,2 6,5
d6 4,2 8,5 1,3 7,6
d7 3,5 8,7 4,6 1,2
Table 1
And Table-2 indicate the transportation cost
between stadiums such as transportation cost
between s1 to s2 is a, s2 to s3 is b, s3 to s4 is c, s1
to s4 is d, s1 to s3 is e and s2 to s4 if f.
Stadiums s1 s2 s3 s4
s1 0 a e d
s2 a 0 b f
s3 e b 0 c
s4 d f c 0
Table 2
Also show by figure1.
a
e
f
d b
c
Fig. 1
And table3 indicate the transportation cost of each
team day by day. D1 is the cost of traversal
s1
s4 s3
s2
2.
ISSN: 2278 – 1323
International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 2, Issue 6, June 2013
2092
www.ijarcet.org
between Stadiums for Team1 at Day1 and Day2.
D2 is the cost of traversal between stadiums for
Team1 at Day2 and Day3 and so on for D3, D4 and
D5.
teams
days
d1 d2 d3 d4 d4 d5
t1 0 d c b b c
t2 b b f c e d
t3 c f 0 a e e
t4 0 c e a a e
t5 a b c 0 f a
t6 a 0 a f 0 c
t7 b f d e c f
t8 c a e e a 0
Table 3
So for the team t1 total transportation cost in the
combination1 of this tournament is TCT1.
TCT1= Transportation Cost of Team t1.
It means
TCT1 = 0 + d + c + b + b + c
=0 *a + 2 * b + 2 * c+ 1 * d + 0 * e + 0 * f
Same for the team t2 total transportation cost in the
combination1 of this tournament is TCT2.
TCT2= Transportation Cost of Team t2
It means
TCT2 = b + b + f + c + e + d
= 0 *a + 2 * b +1* c + 1 * d + 1 * e + 1 * f
And so on for all the remaining teams.
And the total transportation cost of Combination1
is TC1.
Or
If we will take the transportation cost between all
stadiums as an unit cost then transportation cost of
team t1 in Combination1 of this tournament is
TCT1.
TCT1 = 0 + 2 + 2 + 1 + 0 + 0
= 5
TCT2 = 0 + 2 + 1 + 1 + 1 + 1
= 6
TCT3 = 5
TCT4 = 5
TCT5 = 5
TCT6 = 4
TCT7 = 6
TCT8 = 5
And TC1= 41
And for another combination of Hamilton path
tournament we have calculated the total
transportation cost as TC2. Table4 indicate the
another combination of pairs according to Hamilton
path tournament for n=8. If there are 8 team t1, t2,
t3, t4, t5, t6, t7 and t8, 4 stadiums s1, s2, s3 and s4
and 7 days d1, d2, d3, d4, d5, d6 and d7.
Days
Stadium
s1 s2 s3 s4
d1 1,5 2,6 3,7 4,8
d2 1,6 3,4 2,8 5,7
d3 5,4 1,7 6,8 2,3
d4 6,3 7,4 5,2 1,8
d5 7,2 8,3 1,4 6,5
d6 4,2 8,5 7,6 1,3
d7 8,7 1,2 3,5 4,6
Table 4
And table5 indicate the transportation cost of each
team day after previous day. D1 is the cost of
traversal between Stadiums for Team1 at Day1 and
Day2. D2 is the cost of traversal between stadiums
for Team1 at Day2 and Day3 and so on for D3, D4
and D5.
teams
days
D1 D2 D3 D4 D4 D5
t1 0 a f c c f
t2 b c c e 0 a
t3 b f d a f c
t4 f a a b e d
3.
ISSN: 2278 – 1323
International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 2, Issue 6, June 2013
2093
www.ijarcet.org
t5 d d e c f b
t6 a e e d c c
t7 c f 0 a e e
t8 c 0 c f 0 a
So for the team t1 total transportation cost in the
combination1 of this tournament is TCT1.
TCT1= Transportation Cost of Team t1.
It means
TCT1 = 0 + a + f + c + c + f
= 1 *a + 0 * b + 2 * c + 0 * d + 0 * e + 2 *
f
Same for the team t2 total transportation cost in the
combination1 of this tournament is TCT2.
TCT2= Transportation Cost of Team t2
It means
TCT2 = b + c + c + e + 0 + a
= 1 *a + 1 * b + 2 * c + 0 * d + 1 * e + 0 * f
And so on for all the remaining teams.
And the total transportation cost of Combination2
is TC2.
Or
If we will take the transportation cost between all
stadiums as an unit cost then transportation cost of
team t1 in Combination1 of this tournament is
TCT1 and value of TCT1 is
TCT1 = 1 + 0 + 2 + 0 + 0 + 2
= 5
TCT2 = 1 + 1 + 2 + 0 + 1 + 0
= 5
TCT3 = 6
TCT4 = 6
TCT5 = 6
TCT6 = 6
TCT7 = 5
TCT8 = 4
And the total transportation cost of the second
combination of same tournament is TC2.
TC2= 43
And so on for the all remains combinations and
then we select minimum cost and optimized
combination for Hamilton path tournament.
IV ALGORITHM
Optimize_Hamilton_Path_Tournament(A,TC,n, m)
A is the 4 dimension array used to store all the
combination of pairs generated by Hamilton path
tournament and n is the number of team played in
this tournament. m is indicate the number of total
combination and TC used to store the
Transportation cost of combinations.
I. Set p ← 1 min ← MAX_VALUE
Best_Combination ← 1
II. Repeat while p ≤ m
1. Set i ← 1
2. Repeat while i ≤ n
(a) Set sum← 0 D1 ← 0 D2 ←
0 j ← 1
(b) Repeat while j ˂ n
(i) Set k ← 1
(ii) Repeat while k ≤ n/2
(A) If A[j][k][0] = i or A[j][k][1] = i
then do
a. If D1= 0 then set D1 ← k
b. Set D2 ← k
c. If D1 = D2 then
set sum ← sum + 0
Otherwise set sum ← sum + 1
D1 ← D2
d. Go to step (b)
(B) Set k ← k + 1
(iii) Set j ← j + 1
(c) Set TCT[i] ← sum
(d) Set TC[p] ← TC[p] + sum
(e) Set i ← i + 1
3. If min > TC[p] then
set min ← TC[p]
Best_Combination ← p
4. Set p ← p + 1
III. Return Best_Combination
V CONCLUSION
Using Cost constraint on various combinations of
Hamilton path tournament design for n teams, we
have selected the combination of teams and
stadiums such that all teams have to travel
minimum distance for whole tournament. As all the
teams will travel minimum distance, so the cost of
their transportation and other expenses associated
with them will also be minimum. This algorithm
4.
ISSN: 2278 – 1323
International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 2, Issue 6, June 2013
2094
www.ijarcet.org
can be used to design tournaments in a cost
effective manner and also if the teams will travel
less, their performance will also increase. In future,
time constraint can also be added to this algorithm.
REFERENCES
[1] Yoshiko T. Ikebe, Akihisa Tamura: Construction of
Hamilton Path Tournament Designs. Graphs and
Combinatorics (2011) 27:703–711 DOI
10.1007/s00373-010-0998-6
[2] de Werra, D.: Some models of graphs for scheduling
sports competitions. Discrete Appl. Math. 21, 47–65
(1988)
[3] de Werra, D., Ekim, T., Raess, C.: Construction of
sports schedules with multiple venues. Discrete
Appl. Math. 154, 47–58 (2006)
[4] Gelling, E.N., Odeh, R.E.: On 1-factorizations of the
complete graph and the relationship to roundrobin
tournaments, Proc. Third Manitoba Conference on
Numerical Math.,Winnipeg 213–221 (1973)
[5] Ikebe, Y.T., Tamura, A.: On the existence of sports
schedules with multiple venues. Discrete Appl. Math.
156, 1694–1710 (2008) 123 Graphs and
Combinatorics (2011) 27:703–711 711
[6] . Lamken,E.R.,Vanstone, S.A.: Balanced tournament
designs and related topics.Discrete Math. 77, 159–
176 (1989)
[7] Nemhauser, G., Trick,M.: Scheduling a major
college basketball conference. Oper. Res. 46, 1–8
(1998)
[8] Russell, R.A., Urban, T.L.:Aconstraint-programming
approach to themultiple-venue sport-scheduling
problem. Comput. Oper. Res. 33, 1895–1906 (2006)
[9] Schellenberg, P.J., van Rees, G.H.J., Vanstone, S.A.:
The existence of balanced tournament design. Ars
Combinatoria 3, 303–317 (1977)
[10] Urban, T.L., Russell, R.A.: Scheduling sports
competitions on multiple venues. European J. Oper.
Res. 148, 302–311 (2003) 123
Trinetra Kumar Pathak is
pursuing his Master of
Engineering in Software
Engineering from Thapar
University. He has qualified
Gate and is currently
pursuing his six months
thesis on Algorithms.
Vinod Kumar Bhalla is
working as Assistant
Professor in Computer
Science and Engineering
Department, Thapar
University, Patiala since
2001. He is taking courses of
Object Oriented
Programming, Analysis &
Design, Software
Engineering
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